Problem 9 - Algebra 2 and Trigonometry

Question

In \(3-10,\) write the expansion of each binomial. $$ (2 b-1)^{3} $$

Step-by-Step Solution

Answer

The expansion of \((2b - 1)^3\) is \(8b^3 - 12b^2 + 6b - 1\).

Step 1: Apply the Binomial Theorem

The Binomial Theorem states that \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). Here, we choose \(a = 2b\), \(b = -1\), and \(n = 3\). We will expand \((2b - 1)^3\) using this theorem.

Step 2: Calculate the Binomial Coefficients

To expand \((2b - 1)^3\), first calculate the coefficients using \(\binom{3}{k}\) for \(k = 0, 1, 2, 3\). These coefficients are \(\binom{3}{0} = 1\), \(\binom{3}{1} = 3\), \(\binom{3}{2} = 3\), and \(\binom{3}{3} = 1\).

Step 3: Compute Each Term in the Expansion

For each \(k\), calculate the corresponding term in the expansion using the formula \(\binom{3}{k} (2b)^{3-k} (-1)^k\): - When \(k = 0\), the term is \(\binom{3}{0} (2b)^3 (-1)^0 = 1 \times 8b^3 = 8b^3\).- When \(k = 1\), the term is \(\binom{3}{1} (2b)^2 (-1)^1 = 3 \times 4b^2 \times (-1) = -12b^2\).- When \(k = 2\), the term is \(\binom{3}{2} (2b)^1 (-1)^2 = 3 \times 2b \times 1 = 6b\).- When \(k = 3\), the term is \(\binom{3}{3} (2b)^0 (-1)^3 = 1 \times 1 \times (-1) = -1\).

Step 4: Write the Final Expanded Expression

Combine all the terms from Step 3 to write the expanded form of \((2b - 1)^3\) as: \(8b^3 - 12b^2 + 6b - 1\).

Key Concepts

Understanding Binomial Expansion

The binomial expansion is a process where we express a power of a binomial expression, \((a + b)^n\), in an expanded form. This allows us to write expressions that would normally result in lengthy multiplication steps in a more manageable way. For example, consider the expression \((2b - 1)^3\). By using the binomial expansion, we bypass the need to multiply \((2b - 1)\) by itself multiple times.

The Binomial Theorem helps us achieve this. It states that:

• \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\)

This symbolic expression describes exactly how to distribute the powers across \(a\) and \(b\), with the assistance of the binomial coefficients. In the context of our exercise, choosing \(a = 2b\) and \(b = -1\), we can follow the theorem to break down \((2b-1)^3\) into a sum of terms.

This method is especially helpful in algebra as it simplifies the expansion process and helps to avoid errors that might arise from repeatedly multiplying binomials.

Understanding Binomial Coefficients

Binomial coefficients are key components in the binomial expansion. They appear in the expansion formula as \(\binom{n}{k}\). These coefficients tell us how many ways we can pick \(k\) elements from a set of \(n\) elements, often called "n choose k". These coefficients are calculated using the formula:

• \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\)

For the case of \((2b - 1)^3\), the power \(n\) is 3, so we need to find the coefficients for \(k = 0, 1, 2, 3\).

• \(\binom{3}{0} = 1\)
• \(\binom{3}{1} = 3\)
• \(\binom{3}{2} = 3\)
• \(\binom{3}{3} = 1\)

These coefficients help us determine the scaling factor for each term in the binomial expansion. They effectively manage the number of ways each power is distributed in the original expression, transforming our expression in an organized way.

Algebraic Expressions in Binomial Expansion

Algebraic expressions play a fundamental role in the binomial expansion process. When we expand an expression like \((2b - 1)^3\), each term we derive represents an algebraic expression resulting from various powers of parts of the binomial. The terms consist of combinations of coefficients, powers of \(a\), and powers of \(b\).

During the expansion, we must handle each power separately, considering the appropriate binomial coefficient. In our exercise, each term of the expansion was calculated using:

• \(\binom{3}{k} (2b)^{3-k} (-1)^k\)

Here,

• For \(k = 0\), the term is \(8b^3\).
• For \(k = 1\), it is \(-12b^2\).
• For \(k = 2\), it is \(6b\).
• For \(k = 3\), it becomes \(-1\).

The final expanded form of the algebraic expression is a combination of these calculated terms: \(8b^3 - 12b^2 + 6b - 1\). Working through each step ensures that each algebraic expression integrates correctly, providing a comprehensive view of the entire expanded expression.